| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s − 2·5-s + 4·6-s − 4·8-s + 9-s − 4·10-s + 2·11-s + 4·12-s + 16·13-s − 4·15-s − 12·16-s − 10·17-s + 2·18-s − 2·19-s − 4·20-s + 4·22-s + 6·23-s − 8·24-s + 25-s + 32·26-s − 2·27-s + 4·29-s − 8·30-s − 6·31-s − 16·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s − 0.894·5-s + 1.63·6-s − 1.41·8-s + 1/3·9-s − 1.26·10-s + 0.603·11-s + 1.15·12-s + 4.43·13-s − 1.03·15-s − 3·16-s − 2.42·17-s + 0.471·18-s − 0.458·19-s − 0.894·20-s + 0.852·22-s + 1.25·23-s − 1.63·24-s + 1/5·25-s + 6.27·26-s − 0.384·27-s + 0.742·29-s − 1.46·30-s − 1.07·31-s − 2.82·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.330431261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.330431261\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 8 T + 3 p T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 10 T + 44 T^{2} + 220 T^{3} + 1147 T^{4} + 220 p T^{5} + 44 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 23 T^{2} - 22 T^{3} + 292 T^{4} - 22 p T^{5} - 23 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - 16 T^{2} - 36 T^{3} + 1347 T^{4} - 36 p T^{5} - 16 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 18 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 35 T^{2} - 92 T^{3} + 640 T^{4} - 92 p T^{5} - 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 63 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 14 T^{2} - 416 T^{3} - 3653 T^{4} - 416 p T^{5} + 14 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 10 T - 16 T^{2} - 20 T^{3} + 4075 T^{4} - 20 p T^{5} - 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 49 T^{2} + 468 T^{3} - 5112 T^{4} + 468 p T^{5} + 49 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T - 23 T^{2} - 472 T^{3} - 2432 T^{4} - 472 p T^{5} - 23 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 23 T^{2} - 594 T^{3} - 5604 T^{4} - 594 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 4 T^{2} - 828 T^{3} - 9525 T^{4} - 828 p T^{5} - 4 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.48238865047600500967617778295, −9.577915929829843295969819615190, −9.546495171709696014616340960084, −9.099337314521597948990495949845, −8.928652244357981869079406491368, −8.555543927013313073554447965102, −8.530060341764745914046822776734, −8.513907458480985139169851277761, −8.089289450486079690680658953257, −7.28263508513117972600711673571, −7.28193823974852025302801968297, −6.50858148979081882075054991466, −6.49993823231428821026950099325, −6.25473397336995953336814554605, −6.02061208775852973949673648257, −5.75405956470893578673517064721, −4.82032773151661681924888466043, −4.81683772249474723549343217743, −4.32094839033757970734898059383, −3.70199724640457185469268043618, −3.54114164783044621516089240010, −3.42465522315660724767131844910, −3.09315354450759599404135906495, −2.29473201603628499720629466881, −1.60408273106223122148677741851,
1.60408273106223122148677741851, 2.29473201603628499720629466881, 3.09315354450759599404135906495, 3.42465522315660724767131844910, 3.54114164783044621516089240010, 3.70199724640457185469268043618, 4.32094839033757970734898059383, 4.81683772249474723549343217743, 4.82032773151661681924888466043, 5.75405956470893578673517064721, 6.02061208775852973949673648257, 6.25473397336995953336814554605, 6.49993823231428821026950099325, 6.50858148979081882075054991466, 7.28193823974852025302801968297, 7.28263508513117972600711673571, 8.089289450486079690680658953257, 8.513907458480985139169851277761, 8.530060341764745914046822776734, 8.555543927013313073554447965102, 8.928652244357981869079406491368, 9.099337314521597948990495949845, 9.546495171709696014616340960084, 9.577915929829843295969819615190, 10.48238865047600500967617778295
